The exponent $1\mathrm{/}4$ in three complexity bounds: integer bases, continued fractions, and the $(3\mathrm{/}2{)}^n$ steering word

Let $\xi$ be a real algebraic irrational number and $p(n,\xi ,b)$ the number of distinct blocks of length $n$ in its base-$b$ expansion. Bugeaud and Evertse proved in 2008 that $p(n,\xi ,b)\mathrm{/}(n(\log n{)}^{\eta })$ is unbounded for every $\eta <1\mathrm{/}11$. We prove this for every $\eta <1\mathrm{/}4$. We also prove the analogous bound for the continued fraction expansion of an algebraic number of degree at least three, with every exponent $\delta <1\mathrm{/}4$. Bugeaud announced this statement in 2013 with an unspecified $\delta >0$. Finally, the subword complexity of the steering word of $(3\mathrm{/}2{)}^n$ exceeds $Ck(\log k{)}^{\eta }$ infinitely often for every $\eta <1\mathrm{/}4$ and every $C$. The three proofs share one count: ${\epsilon }^{-1}$ classes times ${\epsilon }^{-3}$ subspaces from the quantitative Subspace Theorem of Evertse and Ferretti. The subspaces are then disposed of at the cost of a logarithm, with no further power of ${\epsilon }^{-1}$. All three results, and the quantitative Subspace Theorem they rest on, are machine-verified in Lean~4.